Standard Deviation for Carson Palmer?

Question

I figured a SD for NFL quarterback Carson Palmer the, last 3 years of Total Yards thrown per year (2004, 2005, 2006).
Mean: 3589.33
Standard Deviation: 496.25
This may sound stupid, but:
Can someone explain the 68%, 96%, & 99.7%? Where i'm getting confused is that 1 SD gives a range of 4085.58 to 3093.08, but his highest year was 4035. is the 68, 96,99 a probability or an actual range of the numbers taken to get the mean?

Answer 1

Standard deviation has meaning only for normal or gaussian distribution. For all other distributions it is only partially applicable. For example, for uniform distribution of numbers from 0 to 1 standard deviation is 1/12, so 3 sigma will be 1/4, so you would expect 95% of cases inside 1/2+-1/4, i.e. from 1/4 to 3/4, but you will get only 50%.
It looks that distribution in your case is also far from normal and that is why you see such a discreapency.
In your specific case it looks like he has a "long tail" at values below average and quite "short tail" at value above average. Because standard deviation is "average" deviation in both directions you get standard deviation that is larger then real deviation into values above average, so 3589+497>4035.

Answer 2

Hi,
The numbers 68%, 95%, and 99.7% mean that if the variable is normally distributed (i.e., bell shaped), then 68% of the data will fall within +- one standard deviation of the mean; 95% will fall within +- two standard deviations; and 99.7% within three. Although, this is, strictly speaking, applicable to a normal distribution, in practice it is often applied to any mound-shaped distribution where the number of data points is greater than 30.
This can also be applied as a probability. For example, if a value is randomly selected, there is a probability of .68 that is will fall within +- 1 Standard Deviation of the mean.

Hope this clears up the matter a little.
FE

Answer 3

i think the % are proability .